Cross section, differential oscillations

If we are only interested in the frequency of the modulations in the vicinity of the zero field limit we may employ a different approach, used by Freeman et al,6,7 and Rau10. They used the fact that the motion in the direction is bound and found the energy separation between successive eigenvalues. Specifically, they used Eq. (8.8), the WKB quantization condition for the bound motion in the direction, and differentiated it to find the energy spacing between states of adjacent n1 or, equivalently, between the oscillations observed in the cross sections. Differentiating Eq. (8.8) with respect to energy yields... 

Bandrauk (1969) used a distorted wave Born approximation to calculate the inelastic cross sections of atomic alkali halogen collisions. He found a forwardly peaked differential cross section without oscillations. His total cross section decreases with E 1/2 which is the high energy asymptotic behaviour of the LZ cross section (37). The magnitude of the cross section is much larger than the LZ-result, however. His results were not essentially different if instead of a Coulombic Hl2(R) he used a constant or screened Coulombic interaction. 

Pack R T 1978 Anisotropic potentials and the damping of rainbow and diffraction oscillations in differential cross-sections Chem. Phys. Lett. 55 197... 

The Bom approximation for the differential cross section provides the basis for the interpretation of many experimental observations. The discussion is often couched in temis of the generalized oscillator strength. 

Symmetry oscillations therefore appear in die differential cross sections for femiion-femiion and boson-boson scattering. They originate from the interference between imscattered mcident particles in the forward (0 = 0) direction and backward scattered particles (0 = 7t, 0). A general differential cross section for scattering... 

The corresponding differential cross sections f will therefore exliibit interference oscillations. The integral cross sections are... 

The parameter e0 was chosen for best agreement with the experimental data of Opal et al.52 at = 500 eV. Jain and Khare applied this equation to the calculation of ionization cross sections for C02, CO, HzO, CH4, and NH3 and achieved fairly good agreement with experiment for all cases except for CO, where the cross section was too low, though the ionization efficiency curve still exhibited the correct shape. The main limitation of this method, which it has in common with the BED theory, is the inclusion of the differential oscillator strengths for the target molecule which restricts the number of systems to which it can be applied. 

Figure 11 A Platzman plot, the ratio of the experimental single differential cross section for electron emission from helium by 1-MeV protons to the corresponding Rutherford cross sections plotted as a function of RjE. The experimental cross sections are from Ref. 54 and the differential oscillator strength is taken from Ref. 43.

The relative success of the binary encounter and Bethe theories, and the relatively well established systematic trends observed in the measured differential cross sections for ionization by fast protons, has stimulated the development of models that can extend the range of data for use in various applications. It is clear that the low-energy portion of the secondary electron spectra are related to the optical oscillator strength and that the ejection of fast electrons can be predicted reasonable well by the binary encounter theory. The question is how to merge these two concepts to predict the full spectrum. 

Finally, we bring attention to an experiment by Schepper et al.132 on vibrational excitation of N2 and CO in collisions with ground-state potassium without electronic excitation, who observe a special type of oscillations in the differential cross section. This may be of importance in a detailed discussion of quenching processes in the outgoing channel. 

The differential cross section of a transition into a continuous spectrum can be also expressed in terms of generalized oscillator strengths. However, in this case we must introduce the spectral density of generalized oscillator strengths df q) =/( , g)113 14 ... 

Thus, in order to calculate the differential cross sections it is enough to know the generalized oscillator strengths. On the other hand, if the cross sections are found experimentally, formulas (4.13) and (4.18) enable us to find the experimental values of the oscillator strengths.117 We will briefly dwell on the properties of the generalized oscillator strengths.113,118... 

The binary-encounter-dipole (BED) model of Kim and Rudd [31] couples the modified form of Mott cross section [32] with the Bom-Bethe theory [27]. BED requires the differential continuum oscillator strength (DOS) which is rather difficult to obtain. The simplest approximate version of BED is the binary-encounter-Bethe (BEB) [31] model, which does not need the knowledge of DOS for calculating the EISICS. 

The quasiclassical trajectory method disregards completely the quantum phenomenon of superposition (13,18,19) consequently, the method fails in treating the reaction features connected with the interference effects such as rainbow or Stueckelberg-type oscillations in the state-to-state differential cross sections (13,17,28). When, however, more averaged characteristics are dealt with (then the interference is quenched), the quasiclassical trajectory method turns out to be a relatively universal and powerful theoretical tool. Total cross-sections (detailed rate constants) of a large variety of microscopic systems can be obtained in a semiquantitative agreement with experiment (6). 

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